Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Infinite Time Horizon Optimal Control of McKean-Vlasov SDEs

Silvia Rudà

TL;DR

This work addresses discounted infinite-horizon optimal control for McKean–Vlasov SDEs, establishing well-posedness of the state dynamics and the control problem. It proves a time-invariance property for the value function and shows that the value depends on the initial condition only through its law, allowing a reformulation on the Wasserstein space with a dynamic programming principle and a viscosity solution to an elliptic HJB equation. A key contribution is the Lions derivative-based HJB framework on $\mathcal{P}_2(\mathbb{R}^d)$, together with a finite-horizon approximation technique that yields a uniqueness result under stronger regularity assumptions. The results provide a rigorous analytic foundation for mean-field control in the discounted infinite-horizon setting, with implications for time-consistent planning in large-population systems.

Abstract

We present a theory of optimal control for McKean-Vlasov stochastic differential equations with infinite time horizon and discounted gain functional. We first establish the well-posedness of the state equation and of the associated control problem under suitable hypotheses for the coefficients. We then especially focus on the time invariance property of the value function V, stating that it is in fact independent of the initial time of the dynamics. This property can easily be derived if the class of controls can be restricted, forgetting the past of the Brownian noise, without modifying the value. This result is not trivial in a general McKean-Vlasov case; in fact, we provide a counterexample where changing the class of controls worsens the value. We thus require appropriate continuity assumptions in order to prove the time invariance property. Furthermore, we show that the value function only depends on the initial random condition through its probability distribution. The function V can thus be rewritten as a map v on the Wasserstein space of order 2. After establishing a Dynamic Programming Principle for v, we derive an elliptic Hamilton-Jacobi-Bellman equation, solved by v in the viscosity sense. Finally, using a finite horizon approximation of our optimal control problem, we prove that the aforementioned equation admits a unique viscosity solution under stronger assumptions.

Infinite Time Horizon Optimal Control of McKean-Vlasov SDEs

TL;DR

This work addresses discounted infinite-horizon optimal control for McKean–Vlasov SDEs, establishing well-posedness of the state dynamics and the control problem. It proves a time-invariance property for the value function and shows that the value depends on the initial condition only through its law, allowing a reformulation on the Wasserstein space with a dynamic programming principle and a viscosity solution to an elliptic HJB equation. A key contribution is the Lions derivative-based HJB framework on , together with a finite-horizon approximation technique that yields a uniqueness result under stronger regularity assumptions. The results provide a rigorous analytic foundation for mean-field control in the discounted infinite-horizon setting, with implications for time-consistent planning in large-population systems.

Abstract

We present a theory of optimal control for McKean-Vlasov stochastic differential equations with infinite time horizon and discounted gain functional. We first establish the well-posedness of the state equation and of the associated control problem under suitable hypotheses for the coefficients. We then especially focus on the time invariance property of the value function V, stating that it is in fact independent of the initial time of the dynamics. This property can easily be derived if the class of controls can be restricted, forgetting the past of the Brownian noise, without modifying the value. This result is not trivial in a general McKean-Vlasov case; in fact, we provide a counterexample where changing the class of controls worsens the value. We thus require appropriate continuity assumptions in order to prove the time invariance property. Furthermore, we show that the value function only depends on the initial random condition through its probability distribution. The function V can thus be rewritten as a map v on the Wasserstein space of order 2. After establishing a Dynamic Programming Principle for v, we derive an elliptic Hamilton-Jacobi-Bellman equation, solved by v in the viscosity sense. Finally, using a finite horizon approximation of our optimal control problem, we prove that the aforementioned equation admits a unique viscosity solution under stronger assumptions.

Paper Structure

This paper contains 18 sections, 19 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. McKean-Vlasov SDEs and optimal control
  3. Notation and assumptions
  4. Probabilistic setting
  5. Dynamics of the controlled system
  6. Gain functional and optimization problem
  7. The Time Invariance Property
  8. Preliminary considerations
  9. A counterexample to the time invariance property
  10. Time invariance of the value function
  11. Disintegration of McKean-Vlasov systems
  12. Proof of the time invariance property
  13. Continuity of the Value Function and Law Invariance Property
  14. Hamilton-Jacobi-Bellman equation
  15. Lions Derivative and viscosity solutions to PDEs
  16. ...and 3 more sections

Key Result

Lemma 2.3

Let $\alpha$ be a $\mathbb{F}$-predictable process valued in $A$. Then, there exists a measurable function $\underline{\alpha}: [0, +\infty) \times \mathcal{C}([0,+ \infty); \mathbb{R}^m) \times [0,1] \rightarrow A$ such that, for $\mathbb{P} \text{-a.e. } \omega \in \Omega,$ where $B_{. \wedge s}(\omega)$ denotes the trajectory of the Brownian motion $B$ associated with $\omega$ and stopped at ti

Theorems & Definitions (58)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8: Flow Property
  • Theorem 2.9: Dynamic programming principle
  • ...and 48 more